Question

В классе 10 деталей, из них 8 бракованных. наудачу извлечены 4 детали. найти вероятность того, что среди извлеченных деталей: 1. нет бракованных 2. нет годных

Answer 1

Вероятность того, что будут 4 небракованных детали 0,0016
ВЕроятность того, что все бракованные 0,4096

Answer 2

Problem Analysis

We are given a class of 10 parts, out of which 8 are defective. We randomly select 4 parts from the class. We need to find the probability that among the selected parts: 1. None of them are defective. 2. None of them are usable (non-defective).

Solution

To solve this problem, we can use the concept of combinations. The probability of selecting a specific combination of parts can be calculated by dividing the number of favorable outcomes (i.e., the number of ways to select the desired combination) by the total number of possible outcomes.

Let's calculate the probabilities for each case:

Case 1: None of the selected parts are defective

To calculate the probability of selecting 4 non-defective parts, we need to consider the number of ways to select 4 parts from the 2 non-defective parts out of 10. The total number of possible outcomes is the number of ways to select 4 parts from a class of 10.

The probability can be calculated using the formula: P = (Number of ways to select 4 non-defective parts) / (Total number of ways to select 4 parts)

The number of ways to select 4 non-defective parts is given by the combination formula: C(2, 4) = 2! / (4! * (2-4)!) = 1 / (4 * 3 * 2 * 1) = 1 / 24

The total number of ways to select 4 parts from a class of 10 is given by the combination formula: C(10, 4) = 10! / (4! * (10-4)!) = 10 * 9 * 8 * 7 / (4 * 3 * 2 * 1) = 210

Therefore, the probability of selecting 4 non-defective parts is: P(No defective parts) = (1 / 24) / 210 = 1 / (24 * 210) = 1 / 5040

Case 2: None of the selected parts are usable (non-defective)

To calculate the probability of selecting 4 defective parts, we need to consider the number of ways to select 4 parts from the 8 defective parts out of 10. The total number of possible outcomes is the number of ways to select 4 parts from a class of 10.

The probability can be calculated using the formula: P = (Number of ways to select 4 defective parts) / (Total number of ways to select 4 parts)

The number of ways to select 4 defective parts is given by the combination formula: C(8, 4) = 8! / (4! * (8-4)!) = 8 * 7 * 6 * 5 / (4 * 3 * 2 * 1) = 70

The total number of ways to select 4 parts from a class of 10 is given by the combination formula: C(10, 4) = 10! / (4! * (10-4)!) = 10 * 9 * 8 * 7 / (4 * 3 * 2 * 1) = 210

Therefore, the probability of selecting 4 defective parts is: P(No usable parts) = 70 / 210 = 1 / 3

Summary

The probabilities for the given cases are as follows: 1. The probability of selecting 4 non-defective parts is 1 / 5040. 2. The probability of selecting 4 defective parts is 1 / 3.

Please let me know if I can help you with anything else.